Time allowed: 4 hours Each problem is worth 7 points *The contest problems are to be kept confidential until they are posted on the offi- cial APMO website (http://www.mmjp.or.jp/competi[r]
Trang 12011 APMO PROBLEMS
Time allowed: 4 hours Each problem is worth 7 points
*The contest problems are to be kept confidential until they are posted on the offi-cial APMO website (http://www.mmjp.or.jp/competitions/APMO) Please do not disclose nor discuss the problems over the internet until that date Calculators are not allowed to use
Problem 1 Let a, b, c be positive integers Prove that it is impossible to have all of the three numbers a2+ b + c, b2+ c + a, c2+ a + b to be perfect squares
Problem 2 Five points A1, A2, A3, A4, A5 lie on a plane in such a way that no three among them lie on a same straight line Determine the maximum possible value that the minimum value for the angles ∠AiAjAk can take where i, j, k are distinct integers between 1 and 5
Problem 3 Let ABC be an acute triangle with ∠BAC = 30◦ The internal and external angle bisectors of ∠ABC meet the line AC at B1and B2, respectively, and the internal and external angle bisectors of ∠ACB meet the line AB at C1and C2, respectively Suppose that the circles with diameters B1B2 and C1C2 meet inside the triangle ABC at point P Prove that ∠BP C = 90◦
Problem 4 Let n be a fixed positive odd integer Take m + 2 distinct points
P0, P1, · · · , Pm+1 (where m is a non-negative integer) on the coordinate plane in such a way that the following 3 conditions are satisfied:
(1) P0= (0, 1), Pm+1= (n + 1, n), and for each integer i, 1 ≤ i ≤ m, both x- and y- coordinates of Pi are integers lying in between 1 and n (1 and n inclusive) (2) For each integer i, 0 ≤ i ≤ m, PiPi+1 is parallel to the x-axis if i is even, and
is parallel to the y-axis if i is odd
(3) For each pair i, j with 0 ≤ i < j ≤ m, line segments PiPi+1 and PjPj+1 share
at most 1 point
Determine the maximum possible value that m can take
Problem 5 Determine all functions f : R → R, where R is the set of all real numbers, satisfying the following 2 conditions:
(1) There exists a real number M such that for every real number x, f (x) < M is satisfied
(2) For every pair of real numbers x and y,
f (xf (y)) + yf (x) = xf (y) + f (xy)
is satisfied
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